The present invention relates to fiber optic gyroscopes used for rotation sensing and, more particularly, to interferometric fiber optic gyroscopes.
Fiber optic gyroscopes are an attractive means with which to sense rotation. They can be made quite small and still be constructed to withstand considerable mechanical shock, temperature change, and other environmental extremes. In the absence of moving parts, they can be nearly maintenance free, and they have the potential to become economical in cost. They can also be sensitive to low rotation rates which can be a problem in other types of optical gyroscopes.
There are various forms of optical inertial rotation sensors which use the well-known Sagnac effect to detect rotation about a pertinent axis thereof. These include active optical gyroscopes which have the gain medium contained in an optical cavity therein, such as the ring laser gyroscope, and passive optical gyroscopes without any gain medium in the primary optical path, such as the interferometric fiber optic gyroscope and the ring resonator fiber optic gyroscope. The avoidance of having the active medium along the primary optical path in the gyroscope eliminates some problems which are encountered in active gyroscopes, such as low rotation rate lock-in, bias drift and some causes of scale factor variations.
Interferometric fiber optic gyroscopes typically employ a single spatial mode optical fiber of a substantial length, typically 100 to 2,000 meters, which length is formed into a coil by being wound on a core to form a closed optical path. An electromagnetic wave, or light wave, is introduced and split into a pair of such waves to propagate in opposite directions through the coil to both ultimately impinge on a photodetector. Rotation about the sensing axis of the core, or the coiled optical fiber, provides an effective optical path length increase in one rotational direction and an effective optical path length decrease in the opposite rotational direction for one member of this pair of electromagnetic waves. The opposite result occurs for the remaining member of the pair of electromagnetic waves for such rotation. Such path length differences between the pair of electromagnetic waves introduce a phase shift between those waves in interferometric fiber optic gyroscopes in either rotation direction, i.e. the well-known Sagnac effect. The use of a coiled optical fiber is desirable because the amount of phase difference shift due to rotation, and so the output signal, depends on the length of the entire optical path through the coil traversed by the two opposing directional electromagnetic waves. Thus, a relatively large phase shift difference can be obtained in a long optical fiber, but also in the relatively small volume taken by that fiber in its being coiled.
The output current from the photodetector system photodiode in response to the opposite direction traveling electromagnetic waves impinging thereon, after passing through the coiled optical fiber, follows a raised cosine function, that is, the output current depends on the cosine of the phase difference between these two waves. Since a cosine function is an even function, such an output function gives no indication as to the relative direction of the phase difference shift, and so no indication as to the direction of the rotation about the axis. In addition, the rate of change of a cosine function near zero phase value is very small, and so such an output function provides very low sensitivity for low rotation rates.
Because of these unsatisfactory characteristics, the phase difference between the two electromagnetic waves is usually modulated by placing an optical phase modulator on one side of the coiled optical fiber. As a result, one of the opposite direction propagating waves passes through the modulator just after entering the coil, while the other wave, traversing the coil in the opposite direction, passes through the modulator just before exiting the coil. In addition, a phase sensitive demodulator is provided to receive the photodetector output current. Both the optical phase modulator and the phase sensitive demodulator are typically operated by a sinusoidal signal generator providing a signal of a selected fundamental frequency, but other waveform types of a similar fundamental frequency can also be used. A particularly good choice for this fundamental frequency is the "proper" frequency equal to the value .pi. divided by the propagation delay through the coiled optical fiber, if the system can be operated at that frequency which is usually a relatively high frequency. Operation at this frequency will reduce or eliminate certain phase modulator induced problems such polarization modulation.
The resulting signal output of the phase sensitive demodulator follows a sine function, i.e. the output signal depends on the sine of the phase difference between the two electromagnetic waves impinging on the photodiode, primarily the phase shift due to rotation about the axis of the coil. A sine function is an odd function having its maximum rate of change at zero, and so changes algebraic sign on either side of zero. Hence, the phase sensitive demodulator signal can provide both an indication of which direction a rotation is occurring about the axis of the coil, and the maximum rate of change of signal value as a function of rotation rate near a zero rotation rate. That is, the signal has its maximum sensitivity near zero phase shift so that its output signal is quite sensitive to low rotation rates. This is possible, of course, only if phase shifts due to other sources, that is, errors, are made sufficiently small. In addition, this output signal in these circumstances is very close to being linear at relatively low rotation rates. Such characteristics for the output signal of the phase sensitive demodulator is a substantial improvement over the characteristics of the output current of the photodetector.
Reducing erroneous phase shifts from other sources is, however, a difficult problem in fiber optic gyroscopes. Avoidance of erroneous phase shifts in the electromagnetic waves reaching the photodetector requires that each of the interfering waves, at least those of the same wavelength, have traveled over the same optical path, that is, the electromagnetic wave of a wavelength associated with a clockwise direction of travel from the coil and the one of the same wavelengths associated with the counterclockwise direction of the coil each must travel over an indistinguishable optical path from the source to the photodetector absent any rotation of the coil. A system with this characteristic is often termed "reciprocal." At a minimum, the optical paths corresponding to the common wavelength clockwise electromagnetic waves and counterclockwise electromagnetic waves must be identical on an optical ray tracing basis in the absence of rotation. In meeting this requirement, a "minimum reciprocal configuration" has been found to be as shown in FIG. 1 in connection with the coiled optical fiber, 10, shown there. Coiled optical fiber 10 in FIG. 1 is, as indicated above, wound about a core or spool using a single spatial mode optical fiber wrapped about an axis thereof which becomes the axis about which rotation is to be sensed. The use of such a single mode fiber allows the paths of the electromagnetic waves to be defined nearly uniquely, and further allows the phase fronts of such a guided wave to be defined uniquely. This greatly aids in maintaining reciprocity.
In addition, the optical fiber may be a so-called polarization-maintaining fiber in that a very significant birefringence is constructed in the fiber so that birefringence introduced by mechanical stress, which is unavoidable, and by the Faraday effect in magnetic fields, or from other sources, and which can lead to randomly varying phase difference shifts, becomes relatively insignificant. Thus, either the high refractive index axis, i.e. the slower propagation axis or the "x" axis, or the low refractive index axis, i.e. the faster propagation axis or the "y" axis, is chosen for primarily propagating the electromagnetic waves depending on the other optical components in the system.
On the other hand, such polarization-maintaining optical fiber is relatively expensive so that there is a substantial desire to be able to use just ordinary single spatial mode optical fiber. That desire can be satisfied with the use of primarily ordinary single mode optical fiber for coiled optical fiber 10. However, the optical fiber in coil 10 is then not entirely ordinary single spatial mode optical fiber because of a depolarizer, 10', (having the splices associated therewith shown in dashed lines to indicate that this is an alternative) is included relatively near one end thereof, although this depolarizer could be located anywhere in coil 10. This depolarizer is required because the ordinary single spatial mode optical fiber used in a very great fraction of this version of coil 10 is subject to having changing birefringence therein introduced by mechanical stress changing with temperature, and by the Faraday effect in magnetic fields. This changing birefringence will lead to randomly varying polarization rotations of the beams passing therethrough even to the extent of being so great that the interference of those beams at the photodetector vanishes.
Depolarizer 10' is in coil 10 positioned near one end in coil 10 for ease of winding that coil. Such a depolarizer tends to closely equalize the electromagnetic wave intensities in, and decorrelate, the two orthogonal polarization modes permitted therein and overwhelm the effects of the randomly changing birefringence in the ordinary single spatial mode fiber in the rest of coil 10 thus preventing such opposing direction beam interference at the optical subsystem portion output photodiode from vanishing.
Such a depolarizer can be formed with two lengths of polarization-maintaining fiber, 10" and 10'", with the latter being substantially twice as long as the former to thereby cause approximately twice the optical delay caused by the other. In each of these lengths, there is a high refractive index axis, i.e. the slower propagation axis or the "x" axis, and a low refractive index axis, i.e. the faster propagation axis or "y" axis, which are orthogonal to one another. The lengths are joined in a fused splice in such a manner that the "x" axis in one length is approximately equidistant from the "x" and "y" axes in the other length, i.e. the "x" axis in the former is at 45.degree. from each of the "x" and "y" axes in the other. The opposite ends of each of the depolarization fiber lengths are then spliced by fusing to corresponding portions of the single spatial mode ordinary optical fiber in coil 10 so that a beam of light, propagating through any of the depolarizer or either of the single spatial mode ordinary optical fiber portions, substantially propagates through all of them.
Coil 10, as either polarization-maintaining optical fiber or as ordinary single mode optical fiber with depolarizer 10' therein, is typically wound on a spool using the "quadrupole" technique so that similarly located points in the coil with respect to center are near one another. This reduces the effects of time-varying phenomena, such as thermal gradients, from affecting opposite direction propagating electromagnetic waves differently from one another.
The electromagnetic waves which propagate in opposite directions through coil 10 are provided from an electromagnetic wave source, or light source, 11, in FIG. 1. This source is typically a superluminescent diode or, alternatively, a laser diode operating below its threshold for stimulated emission, either of which provide electromagnetic waves typically in the near-infrared part of the spectrum with a typical wavelength of 1.3 .mu.m. Source 11 must have a short coherence length for emitted light to reduce the phase shift difference errors between these waves due to Rayleigh scattering at scattering sites in coil 10. Because of the nonlinear Kerr effect in coil 10, differing intensities in the two propagating waves can also lead to phase difference shifts therebetween. This situation can also be aided by the use of a short coherence length source for source 11 which leads to modal phase shift canceling. Rayleigh scattering and the nonlinear Kerr effect lead to non-reciprocal phase shifts between the counter rotating electromagnetic waves in coil 10 even in a minimum reciprocal configuration. A superluminescent diode, or a laser diode operating below threshold, each have a wide emission spectrum compared to that of a laser diode operating past its threshold in the stimulated emission mode of operation. In addition, such diodes introduce intensity noise into the system of FIG. 1 leading to a source of error in the output signal as will be further described below.
Between laser diode 11 and fiber optic coil 10 in FIG. 1 there is shown an optical path arrangement formed by an extension of the ends of the optical fiber forming coil 10 to some optical coupling components which separate the overall optical path into several optical path portions. A portion of polarization-maintaining or ordinary single spatial mode optical fiber is positioned against a face of laser diode 11 at a location of optimum light emission therefrom, a point from which it extends to a first optical directional coupler, 12, to be joined thereto. If, on the other hand, coupler 12 is formed by fusing two optical fibers together in a coupling region, either a pair of polarization-maintaining optical fibers or a pair of ordinary single spatial mode optical fibers, the excess length of one of the optical fibers may be positioned against diode 11 to provide this optical path between diode 11 and this wave coupling region of coupler 12, or the excess length may be spliced to another polarization-maintaining optical fiber or ordinary single spatial mode optical fiber extending from diode 11 depending, in either of these situations, on which of the coil 10 configurations described above is chosen or the choice of variations of systems having one of those configurations therein.
Optical directional coupler 12 has light transmission media therein which extend between four ports, two on each end of that media, and which are provided on each end of coupler 12 in FIG. 1. One of these ports has the optical fiber extending from laser diode 11 positioned thereagainst (or vice versa for a fused coupler, i.e. a fiber extending from the coupler coupling region to be positioned against the emitting face of diode 11). At the other port on the same end of optical coupler 12 there is shown a further optical fiber positioned thereagainst (or alternatively extending from the fused coupler if used) which extends to be positioned against a photodiode, 13, which is electrically connected to a photodetection system, 14, providing the operating circuitry therefor. This optical fiber may be a polarization-maintaining optical fiber or it may be an ordinary single spatial mode optical fiber. In practice, as indicated above, coupler 12 may be formed from fused lengths of such optical fiber so that the remaining lengths past the fused portion, or the light coupling region therein, extend either all the way to laser diode 11 and photodiode 13, or are spliced to other optical fibers extending therefrom.
Photodiode 13 detects electromagnetic waves, or light waves, impinging thereon from the portion of the optical fiber positioned thereagainst (or extending thereto) and provides a photocurrent in response. This photocurrent, as indicated above, in the situation of two nearly coherent electromagnetic waves impinging thereon, follows a raised cosine function in providing a photocurrent output which depends on the cosine of the phase difference between such a pair of electromagnetic waves, as will be shown below. Photodiode 13 is operated in either the photovoltaic mode or the photoconductive mode, as needed, into an amplifier circuit of appropriate impedance to provide a photocurrent which is substantially a linear function of the impinging radiation intensity, so that it will also have a component therein due to intensity noise emanating from source 11. Typically, photodiode 13 is a p-i-n photodiode.
Optical directional coupler 12 has another optical fiber against a port at the other end thereof which extends to a polarizer, 15. This may be polarization-maintaining or a single spatial mode optical fiber depending on choices of coil 10 configurations and system variations having one or the other of those configurations. At the other port on that same side of coupler 12 there is a non-reflective termination arrangement, 16, involving the excess length of one of the optical fibers fused together forming coupler 12 or, again, another optical fiber spliced to such an excess length. This optical fiber leading to arrangement 16 can again be polarization-maintaining optical fiber or ordinary single spatial mode optical fiber.
Directional optical coupler 12, in receiving electromagnetic waves, or light, at any port, or at any end of an excess portion of optical fiber extending past the coupling region therein, transmits such electromagnetic waves so that a preselected fraction thereof, typically one-half, appears at each of the two ports, or ends of the two excess optical fiber lengths past the coupling region, which are at the opposite end of coupler 12 from that having the incoming port or excess optical fiber length receiving the incoming waves. On the other hand, no electromagnetic waves are transmitted to the port or excess fiber length which is on the same end of coupler 12 as is the incoming port. The polarization of the incoming electromagnetic waves with respect to the principal refringent axes at the input port can be fairly well preserved at the corresponding axes of the two output ports if coupler 12 is formed of two portions of polarization-maintaining optical fiber with the principal axes suitably aligned, but there will be some coupling of waves between axes in the coupling region of the coupler. If a pair of ordinary single spatial mode optical fiber portions are fused together to form coupler 12, the polarization of the incoming electromagnetic waves with respect to the principal birefringent axes in the polarization-maintaining component can be fairly well preserved through the coupling region to the other fiber, but there may be substantial coupling thereafter even before coupled waves reach the output port of the ordinary single spatial mode optical fiber.
Polarizer 15 is used because, even in a single spatial mode optical fiber, two polarization modes are possible for electromagnetic waves passing through such a fiber along orthogonal axes. Thus, polarizer 15 is provided for the purpose of transmitting the electromagnetic wave component along one of these axes, for one of these polarization modes, between the optical fibers connected to the ports on either end thereof. At the same time, polarizer 15 substantially blocks transmission along the remaining one of these axes. Polarizer 15, however, is not capable of entirely blocking electromagnetic waves in the one state of polarization that it is intended to block. This shortcoming in the extinction coefficient thereof leads to a non-reciprocity between two opposite direction traveling waves over the optical paths they follow, and so to a non-reciprocal phase shift occurring between them which can vary with the conditions of the environment in which the polarizer and the remainder of the system of placed.
Positioned against the port of polarizer 15 on the end opposite that connected with optical directional coupler 12 is another optical fiber which extends to a further optical directional coupler, 17, this fiber and this coupler being formed of pairs of polarization-maintaining or ordinary single spatial mode fibers again depending on the choices of coil 10 configurations and system variations having one of these configurations therein. Directional coupler 17 also transmits received electromagnetic waves so that a preselected fraction thereof, again typically one-half, appears at each of the two ports which are at the opposite end of coupler 17 from that having the incoming port. Again, no electromagnetic waves are transmitted to the port or excess fiber length which is on the same end of coupler 17 as the incoming port. The polarization of incoming electromagnetic waves at an input port will be preserved at the corresponding pair of output ports to the extent and manner described for coupler 12. If directional coupler 17 is formed using a pair of portions of polarization-maintaining optical fiber, this will lead to an optical performance in the optical subsystem portion of FIG. 1 which would be similar to the performance of such a subsystem if directional coupler 17 was alternatively formed in an integrated optic chip.
The second port on the same end of coupler 17 from which the first port is coupled to polarizer 15 is connected in a non-reflective termination arrangement, 18, using a further ordinary single spatial mode optical fiber portion or a polarization-maintaining optical fiber. One of the ports on the opposite end of coupler 17 is connected to a further optical component in the optical path portion extending thereto from one end of the optical fiber in coil 10. The other port on that end of coupler 17 is directly coupled to the remaining end of optical fiber coil 10, and this coupling is typically accomplished through a splice between the excess length of an optical fiber past the coupling region in coupler 17 and the optical fiber in coil 10.
Between coil 10 and coupler 17, on the side of coil 10 opposite the directly connected side thereof, there is provided an optical phase modulator, 19. Optical phase modulator 19 has a port on either end of the transmission media contained therein which occur in FIG. 1 at the opposite ends of that phase modulator. The polarization-maintaining or ordinary single spatial mode optical fiber from coil 10 is positioned against a port of modulator 19. The polarization-maintaining or ordinary single spatial mode optical fiber extending from coupler 17 is positioned against the port on the opposite end of modulator 19.
Optical phase modulator 19 can be of the variety formed by wrapping an optical fiber portion around a piezoelectric cylinder so that the fiber may be stretched by the application of voltage to that cylinder, or this phase modulator may be formed as an optical integrated chip using a substrate of lithium niobate, for instance, with metallic depositions provided thereon as electrodes and positioned adjacent a waveguide provided therein. Such depositions typically result in plate-like electrode structures on the substrate to both provide electrical contacts to the modulator and a means through which varying electric fields can be established in the waveguide to result in the necessary modulation of the phase of electromagnetic waves passing through that waveguide.
Optical phase modulator 19 is thus capable of receiving electrical signals on these plates to cause the introduction of phase differences in electromagnetic waves transmitted therethrough by changing the index of refraction of the transmission medium, or transmission media, because of the resulting electric fields established therein to thereby change the effect of optical path lengths experienced by such waves. Optical phase modulators constructed in optical integrated circuit form have a large bandwidth, i.e. are able to provide phase changes following a waveform that has substantial high frequency content. Note also that polarizer 15, and source and loop optical directional couplers 12 and 17, could also be formed in similar integrated optic chips, including possibly being formed in a common such chip.
Directional optical coupler 17 serves as a beam-splitting apparatus in that electromagnetic waves emitted from source 11 that are transmitted through coupler 12 and polarizer 15 to be received by coupler 17 are there split in approximately half with a corresponding one of the resulting portions passing out of each of the two ports on the opposite end of coupler 17. Out of one port on that opposite end of coupler 17 the corresponding electromagnetic wave portion passes through depolarizer 10' if used, the rest of optical fiber coil 10, through optical phase modulator 19 and back to coupler 17. A portion of that electromagnetic wave passes through the port of coupler 17 leading to polarizer 15 and then to coupler 12 where a part of the remainder of the wave portion is transmitted to photodiode 13.
The other portion of the electromagnetic wave after the split in coupler 17 leaves that other port on the coil 10 end of coupler 17 to first pass through optical phase modulator 19, through most of optical fiber coil 10, and then through depolarizer 10' if used to reenter coupler 17 and, again, from there follow the same path as the first portion previously described to finally impinge in part on photodiode 13. In the presence of modulation provided by phase modulator 19, and in the presence of any rotation of coil 10 about its axis, or because of effects in coupler 17, some of the energy of the combined waves will be lost through non-reflective arrangement 18.
In an interferometric fiber optic gyroscope using polarization-maintaining optical fiber for coil 10 without a depolarizer, the electromagnetic waves passing through coil 10 are all intended to take the same optical path. In an interferometric fiber optic gyroscope using ordinary single spatial mode optical fiber for coil 10 with a depolarizer, however, the nature of the ordinary single spatial mode optical fiber used in coil 10 gives rise to random occurrences of birefringence therein induced by various causes, including stress change due to temperature changes, which result in the possibility of different optical paths being available for the waves to propagate over. The use of depolarizer 10' forces waves to differing polarization states periodically over wavelength, and so to corresponding different optical paths. Thus, the polarization history of electromagnetic waves through coil 10 and depolarizer 10' together is wavelength dependent. Nevertheless, any waves reaching the transmission axis of polarizer 15 at a point in time will have had the same polarization history. Assuming then that depolarizer 10' distributes the optical waves between the polarization states uniformly, depolarizer 10' acts to equalize the wave energy in each of the optical paths.
The choice in an interferometric gyroscope system of which of the configurations describe above for coil 10 to use will depend on many factors, as will the choice of system variations having one or the other of the coil configurations therein. A number of different system configurations for each of the coil configurations (as well as variations of those configurations) can be found in the earlier filed copending U.S. patent applications entitled "Configuration Control of Mode Coupling Errors" having Ser. No. 07/791,719 by J. Blake and J. Feth and "Configuration Control of Mode Coupling Errors" having Ser. No. 07/890/938 by J. Blake, J. Feth and B. Szafraniec each hereby incorporated herein by reference.
As indicated above, photodiode 13 provides an output current proportional to the intensity of the combined electromagnetic waves, or light waves, impinging thereon dependent on the phase difference therebetween. The arrangement of FIG. 1 leads to the electromagnetic waves propagating in opposite directions through coil 10 to in part reach photodiode 13 so that the intensity thereon is an average of the electromagnetic waves traveling in both directions over each polarization determined optical path, i.e. averaged over the wavelengths present, but including primarily only those waves propagating over those optical paths over which returning waves have a polarization at polarizer 15 which is substantially passed by that polarizer. That is, the returning waves included in the averaging process are primarily just those following optical paths which extend through the transmission axis of polarizer 15. Corresponding photocurrent from photodiode 13 follows a raised cosine function in being based on the cosine of the average phase difference between portions of each of the electromagnetic waves propagating in opposite directions in coil 10 impinging thereon taken over the wavelengths present therein. This relationship follows because the photocurrent depends on the resulting optical intensity of the pairs of opposite direction propagating electromagnetic waves incident on photodiode 13 which intensity will vary depending on how much constructive or destructive interference occurs between these waves at that diode. This interference of waves will change with rotation of the coiled optical fiber forming coil 10 about its axis as such rotation introduces a phase difference shift between the waves because of the Sagnac effect. Further, additional phase difference shifts will be introduced by optical phase modulator 19 as will be described in connection with the electrical system shown in the remainder of FIG. 1.
This situation can be shown for the system of FIG. 1 by considering in a general overview manner the clockwise and counterclockwise electromagnetic waves which propagate through that system from source 11 to photodetector 13. The waves will be considered to propagate through the system of FIG. 1 assuming that it is constructed using polarization-maintaining optical fiber without the presence of a depolarizer. In addition, common error sources such as due to different polarizations being present in the system because of the imperfection of polarizer 15 in eliminating the waves traveling along the faster propagation axis of the optical fibers by their being aligned with the blocking axis of that polarizer, backscattering at interfaces between different optical components in the system, nonlinearities in the system such as occur in the structure of phase modulator 19 or such as due to the Kerr effect in the presence of unequal intensities in the clockwise and counterclockwise waves, and the like will be assumed negligible or to have otherwise been alleviated to become insignificant in the operation of the system. One such system, for example, would be a system constructed entirely of polarization-maintaining optical fiber with a source that is sufficiently broadband in its emissions, and perhaps having intensity fluctuations of such a nature as to reduce the Kerr effect.
The electric field component of the clockwise propagating electromagnetic wave reaching photodiode 13 in photodetection system 14, E.sub.d-cw, can be represented as ##EQU1## and the counterclockwise propagating wave component, E.sub.d-ccw, as ##EQU2## Here, E.sub.i represents the input electromagnetic electric field component from source 11 of the selected polarization with .tau. representing the elapsed time from leaving source 11 to reaching photodiode 13. Assuming that couplers 12 and 17 have a transfer ratio of one-half the intensity of the incoming wave, there will be a loss of 1/.sqroot.2 of the electromagnetic wave electric field component at each of couplers 12 and 17 so as to result in a loss of one-half of the field component in the complete trip through the system of FIG. 1 from source 11 to photodiode 13. Other losses will occur for each of the waves in passing through the system of FIG. 1 which will be essentially for each of the waves because of the "minimum reciprocal" configuration used in FIG. 1 assuring the same optical paths for both the clockwise and counterclockwise waves. These other losses are represented for each wave by .sqroot.L.
The phases of the waves are represented in the complex exponents of the corresponding exponential factors in these equations. Each of the waves experiences half of the phase change, .phi..sub.R, due to the Sagnac effect during rotation of the system of FIG. 1 about the symmetrical axis of coil 10 oriented perpendicular to the plane of the paper of the figure, but of opposite sign, since rotation will be in the same direction as one of the propagating electromagnetic waves but in the opposite direction to the other. The phase modulation depth, .phi.'.sub.m, is the same for each of the waves, but occurs later for the counterclockwise wave by time .tau. representing the wave transit time through coil 10 which is very close to the transit time from source 11 to photodiode 13, and so the same time is used in the sinusoid in the exponent as was used in the argument of the input radiation from source 11.
The intensity of an electromagnetic wave is, as is well known from electromagnetic theory, equal the square of the electric field component of that wave, and so the intensity of electromagnetic waves emitted by source 11 is equal to I.sub.i =E.sub.i.sup.2. The electric field components of these magnetic waves from source 11 propagate through the system of FIG. 1 as E.sub.cw and E.sub.ccw to reach photodiode 13 of photodetection system 14 where they are additively combined so that the intensity of the electromagnetic waves impinging on photodetector 13, I.sub.D (t), is obtained from the summation of these waves, or EQU I.sub.D (t)=.vertline.E.sub.ccw +E.sub.cw .vertline..sup.2.
Substituting for the expressions found for electric field components of the clockwise and counterclockwise waves into this last equation and substituting the intensity of the electromagnetic waves from source 11 yields ##EQU3## which, with use of the well-known Euler formula and the definition of absolute value from complex variables theory and trigonometric identities, becomes ##EQU4##
Considering just the difference in the two sine functions in the last equation, use of the substitution t t'+.tau./2 and trigonometric identities, permits converting this difference in functions to EQU sin .omega..sub.m t-sin .omega..sub.m (t-.tau.)=2 sin .omega..sub.m .tau./2 cos .omega..sub.m (t-.tau./2).
Use of this last equation in the preceding equation yields ##EQU5## Defining .phi..sub.m 2.phi.'.sub.m sin .omega..sub.m .tau./2 then gives ##EQU6## This last relationship then gives the general performance of the optical subsystem in FIG. 1 as was described above.
The remaining electrical subsystem portion of FIG. 1 shows an open loop fiber optic gyroscope system, but could also be converted to a closed loop fiber optic gyroscope system, i.e. using feedback around the system shown. This would be accomplished by having the electrical system provide a feedback signal based on the output of the system shown in FIG. 1 to control a further optical phase modulator inserted in the optical path next to modulator 19, or to additionally control modulator 19. Optical phase modulator 19 is of the kind described above and is used in conjunction with a phase sensitive demodulator, or phase detector, for converting the output signal of photodiode 13 in photodetection system 14, following a raised cosine function as indicated in the last expression contained above, to a signal following a sine function which is obtained in the demodulation process from this last expression above. Following such a sine function provides, in that output signal, information both as to the rate of rotation and as to the direction of that rotation about the axis of coil 10. Modulator 19 is operated by a sinusoidal signal provided at the output of a bias modulation signal generator, 20, which also provides this signal to operate a phase detector which, as indicated, is a phase sensitive demodulator.
Thus, the output signal from photodetection system 14, including photodiode 13, is provided to an amplifier, 21, where it is amplified and passed through a filter, 22, to a phase detector, 23. The phase sensitive demodulator serving as phase detector 23 is a well-known device. Such a phase sensitive demodulator senses changes in the first harmonic, or fundamental frequency, of signal generator 20 to provide an indication of the relative phase of the pair of electromagnetic waves impinging on photodiode 13. This information is presented by phase detector 23 in an output signal following a sine function, i.e. the sine of the phase difference between the two electromagnetic wave portions impinging on photodiode 13.
Such an output signal can be found from expanding the last expression in a Bessel function series to thereby display the harmonics present in the optical subsystem output signal I.sub.D. Such a series expansion gives ##EQU7## After conversion to an electrical output signal in photodetector system 14, this output signal must have the portion corresponding to the harmonic selected therefrom. Filter 22 is needed because bias modulation signal generator 22 also generates a strong second harmonic component, sensed by photodetection system 14, which needs to be eliminated. Filter 22 passes primarily the first harmonic frequency component from the last equation, i.e. the modulation frequency component, varying at radian frequency .omega..sub.m. As a result, the output signal of filter 22 is ##EQU8## where k.sub.o is the system gain constant arising because of the passage of the signal through photodetection system 14, amplifier 21 and filter 22. A further phase delay term may be added as a result of passing through this amplifier, although this will be ignored, but may be considered to be combined in the phase term shown for this cosinusoid, -.omega..sub.m .tau./2.
This signal from filter 22 is then applied to phase-sensitive detector 23, as is the signal from bias modulator generator 20, the latter being equal to a sinusoid sin .omega..sub.m t at some amplitude. The output of phase-sensitive detector 23 will then be EQU v.sub.23 =-k'.sub.o LI.sub.i (t-.tau.)J.sub.1 (.phi..sub.m) sin .phi..sub.R,
where the constant k'.sub.o accounts for the further system gain resulting from the filter output signal passing through phase-sensitive detector 23.
In operation, the phase difference changes in the two opposite direction propagating electromagnetic waves passing through coil 10 in the optical paths therethrough to reach photodiode 13 will lead to average net phase difference changes which will be relatively small, and which will vary relatively slowly compared to the phase difference changes due to optical phase modulator 19 and bias modulator signal generator 20. Any average phase difference shift due to the Sagnac effect will merely shift the average phase difference between the electromagnetic waves, and the output signal from phase sensitive demodulator 23, after photodiode signal demodulation therein, will depend on the sine of this phase difference multiplied by an amplitude scaling factor set by the modulation of the waves due to phase modulator 19 and signal generator 20. This synchronous demodulation thus substantially extracts from the photodiode output signal the amplitude of the sinusoidal modulation frequency component at the modulation frequency introduced by signal generator 20 and modulator 19, which includes the result of any rotation of coil 10 about its axis, to provide the demodulator output signal.
As indicated above, however, additional phase shifts between the counter-propagating electromagnetic waves can be introduced even with the fiber optic gyroscope system in a minimum reciprocal configuration by various effects occurring therein. Typically, a significant source of such non-reciprocal phase shifts from other than the Sagnac effect is the following of different optical paths by the two different polarization components of the counter-propagating electromagnetic waves because of polarizer 15 being imperfect leading to phase shift errors in the output being distinguishable from the Sagnac phase shifts as described in the references incorporated above. Other sources are backscattering in the optical path and intensity dependence of the index of refraction along the optical paths. As was done above in the finding of the general response of the system of FIG. 1, these sources of errors will be considered either negligible or otherwise alleviated to render them negligible hereafter.
Another source of variation in the phase shifts between the counter-propagating electromagnetic waves in coil 10 leading to gyroscope output errors is due to the variation in the amplitude of the electromagnetic waves emitted by source 11 due to optical noise occurring therein. Even in the absence of noise from source 11, photodetection system 14 with photodiode 13 therein is subject to errors in the output photocurrent due to noise sources contributing thereto. In addition to the signal current, there will be present dark current noise and possible noise due to background radiation reaching photodiode 13. In addition, there will be shot noise due to the statistical nature of the photodetection process and there will be thermal noise generated in the load resistance equivalent present between the two leads of photodiode 13. Thus, in addition to these noise source currents already present in the output signal from photodetection system 14, there will be added contributions due to the optical noise generated by source 11.
There are many sources of light intensity fluctuation in the various kinds of source devices which can be used for optical source 11. All these noise sources combine to increase the detected noise above the limit set by the minimum amount of noise generated independently in photodetection system 14. This excess noise, often termed relative intensity noise, includes flicker or 1/f noise, current noise generated in the source operating electrical circuitry, and carrier density fluctuations. The intensity of the electromagnetic waves emitted by source 11 can be represented in these circumstances as EQU I.sub.i (t)=I.sub.o +n(t),
where I.sub.o is a desired, and substantially constant, electromagnetic wave intensity which source 11 is operated to provide, and n(t) is the noise power which is a random variable that is the result of the contributions of the stochastic processes each noise source represents.
As higher power levels of thermal broadband light are obtained from the optical devices suitable as optical source 11, and at higher operating frequencies where 1/f noise is sufficiently diminished, another fundamental noise source begins to dominate. This is intensity noise that arises in a broadband electromagnetic wave source because of neighboring optical emitters in the source emitting optical frequency electromagnetic waves that mix with one another to leave relatively low frequency intensity fluctuations in the emitted electromagnetic waves from that source. That is, a composite electromagnetic wave arises from these emissions at any instant of time that is the sum of many independent amplified spontaneous emission events. This composite wave can be viewed as a random phasor sum of the many emitted waves since each can have a complex-value representation as a "phasor" and the relative phases of each are uncorrelated, and so the composite forms what is commonly termed as "thermal" light. Consequently, the relative phases of the various spectral components are also uncorrelated. Thus, the intensity fluctuates in time and the various relatively low frequency components of these intensity fluctuations are again uncorrelated. Such intensity noise is common from light-emitting diodes which emit at the edge thereof, and from superluminescent diodes. Laser diodes operated below threshold to obtain greater line widths than occur in stimulated emission will also exhibit intensity noise that can dominate if the operating point is not far below threshold.
The various emitters in the optical source, in each providing an output wave that beats against the others, can each be represented as a phaser with an amplitude and phase of a monochromatic or nearly monochromatic wave disturbance. The amplitude and phase of each can usually be reasonably considered statistically independent of one another and of each of the amplitudes and phases of the other phasers, but with common probability distributions for the amplitudes and phases of each.
The complex addition of the contributions of these small independent phasers provides the representation of the output of the optical source as a whole, and the result is that the source appears to emit white light in a zero mean process in addition to the desired output intensity, at least at sufficiently low frequencies, with the root-mean-square (rms) noise spectrum being given by ##EQU9## where .tau..sub.c is the source coherence time. Since the coherence time of the source decreases as the spectral bandwidth thereof increases, increases in the source spectral bandwidth will decrease this noise. The source coherence time, as is well known, and obtained from ##EQU10## where .gamma.(.tau.) is the source complex degree of coherence function. The noise power n(f) for rms averaging is related to the Fourier transformer of n(t).
As can be seen from the expression given above for the rms noise power, the noise is proportional to source intensities so that increasing the intensity will not improve the signal-to-noise ratio for the system of FIG. 1. Hence, beyond a certain intensity, typically about 10 .mu.w where the intensity noise begins to dominate, there ceases to be any significant improvement in the signal-to-noise ratio of the system of FIG. 1 as the optical intensity of the electromagnetic waves emitted by source 11 is increased.
Since the system of FIG. 1 uses the first harmonic from the optical signal varying at a radiant frequency .omega..sub.m that is obtained from photodetection system 14 as the output signal for the system, noise signal contributions at that frequency, and frequencies nearby, will be especially important in their effects on the output signal obtained from the system. Since the output signal will only be considered within a certain bandwidth by systems in which the system of FIG. 1 is installed, set either by filters or by sampling in versions from analog-to-digital signals, or by the action of filter 22 or other band limiting actions occurring in the system of FIG. 1, the noise signal contributions of significance will only be within a relatively narrow band about the first harmonic signal frequency.
The application of white noise of zero mean applied to the input of a narrow bandwidth filter provides an output signal related to the square of the absolute value of the transfer function of the filter in the frequency domain, as is well known. Through use of the Hilbert transform, the corresponding analytic signal can be formed and through two further manipulations, too involved to be shown here, a representation of the narrow band noise at the output of the filter having a passband centered around .omega. can be provided in terms of envelope and phase components as EQU .delta.I(.omega.) cos [.omega.t+.phi.(.omega.)].
The noise signal envelope is represented as the random variable .delta.I(.omega.) and the phase component as the random variable .phi.(.omega.). The probability distributions of these random variables, including the envelope random variable with a signal present, can be found analytically assuming the noise process is Gaussian and assuming each of these random variables to be varying at frequencies much less than the modulation frequency. These two random variables also vary slowly in time in moving form one random value to another because of the relatively long time constant resulting from the relatively narrow bandwidth of the passband. As a result, source intensity I.sub.i is now represented as EQU I.sub.i (t)=I.sub.o .delta.I.sub.(.omega.) cos [.omega.t+.phi.(.omega.)].
In these circumstances, the output intensity I.sub.D (t) of the optical subsystem portion of the system of FIG. 1, after substituting this last expression into the expression found above for that output signal, becomes ##EQU11## Clearly, a noise component result is added to the output signal of the optical subsystem and, as indicated above, increasing the desired intensity of emitted waves I.sub.o from optical source 11 does not change the signal-to-noise ratio of the system of FIG. 1. Thus, an arrangement is desired for use in connection with the system of FIG. 1 to reduce the effects of the optical intensity noise emanating from the optical source of that system but an arrangement that does not impose too significant a cost.